How would one find the secret key in a simple RSA encryption when given p, q and e?
You've already been given everything you need to decrypt any messages.
RSA key generation works by computing:
- n = pq
- φ = (p-1)(q-1)
- d = (1/e) mod φ
So given p, q, you can compute n and φ trivially via multiplication. From e and φ you can compute d, which is the secret key exponent. From there, your public key is [n, e] and your private key is [d, p, q]. Once you know those, you have the keys and can decrypt any messages - no cracking necessary!
More details are available here.